Quantum modular forms were defined in 2010 by Zagier; they are somewhat analogous to ordinary modular forms, but they are defined on the rational numbers Q as opposed to the upper half complex plane H, and have modified transformation properties.  In 2016, Bringmann and the author defined the notion of a quantum Jacobi form, naturally marrying the concept of a quantum modular form with that of a Jacobi form (the theory of which was developed by Eichler and Zagier in the 1980s).  We will discuss these intertwined topics, emphasizing recent developments and applications.  In particular, we will discuss applications to combinatorics, topology (torus knots), and representation theory (VOAs).